(a) The electrical power demand of TATA Power Corporation from 2018 to 2024, measured in megawatts, is provided below. Using the data, forecast the demand for the year 2025 by fitting and plotting the linear trend line.

Year | Electrical Power Deman

Question

(a) The electrical power demand of TATA Power Corporation from 2018 to 2024, measured in megawatts, is provided below. Using the data, forecast the demand for the year 2025 by fitting and plotting the linear trend line.

Year | Electrical Power Demand (Megawatts)
2018 | 94
2019 | 99
2020 | 100
2021 | 110
2022 | 125
2023 | 162
2024 | 142

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(b) In a study to evaluate the effectiveness of two different animal feeds, researchers recorded the weight gains (in kilograms) of animals after fed each type of feed.

Feed A | 45 | 48 | 47 | 50 | 46 | 49 | 47 | 48
Feed B | 51 | 53 | 52 | 55 | 50 | 52 | 54 | 33

(i) If the two samples are considered independent, can we conclude that Feed B is more effective than Feed A at the 0·025 level of significance ?

(ii) Also examine the case if the same set of eight animals received both feeds, and test at the 0·01 level of significance. (Table enclosed)
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(c) Techline Pvt. Ltd. is a consumer electrics manufacturing concern producing three models of Smart Devices : X, Y and Z. The company operates under the constraints of two key resources, viz. raw materials and labor hours. Each week, the availability of raw materials is restricted to 800 kg, while labor availability is limited to 600 hours. To maximize profit, the company needs to decide, how many units of each product to manufacture weekly without exceeding the available resources. The relevant details for each product are provided below :

| Product | Raw Material per unit (Kg) | Labor Hours per unit (Hours) | Profit contribution per unit (₹) |
|---------|---------------------------|------------------------------|----------------------------------|
| X | 6 | 5 | 80 |
| Y | 4 | 4 | 60 |
| Z | 2 | 3 | 40 |

(i) Formulate this scenario as a Linear Programming Problem (LPP), and solve it for optimal product mix.

(ii) Based on the optimal solution, analyze the situation if the company wants to introduce a New product W, requiring 2 kg of raw materials and 2 hours of labor hours per unit. Its estimated profit contribution is ₹35/- per unit.

Should the company include this new product in its production mix ?
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UPSC Answer Check · Accepted Answer

Solve this multi-part numerical question by allocating approximately 30% time to part (a) trend analysis with linear regression, 30% to part (b) hypothesis testing for both independent and paired samples, and 40% to part (c) linear programming formulation and sensitivity analysis. Begin with clear problem identification, show all computational steps with formulas, present results in structured tables, and conclude with managerial interpretations for each sub-part.
- Part (a): Calculate linear trend line using least squares method (Y = a + bX), compute 2025 forecast, and describe plotting procedure with proper axes labeling
- Part (b)(i): Conduct independent samples t-test with pooled variance, calculate t-statistic, compare with critical t-value at α=0.025 (one-tailed), and state conclusion about Feed B effectiveness
- Part (b)(ii): Perform paired t-test for same animals, compute mean difference and standard error, compare with critical t-value at α=0.01, and interpret results
- Part (c)(i): Formulate LPP with objective function (Max Z = 80X + 60Y + 40Z) and constraints (6X+4Y+2Z ≤ 800, 5X+4Y+3Z ≤ 600, X,Y,Z ≥ 0), solve using graphical or simplex method, identify optimal product mix and maximum profit
- Part (c)(ii): Calculate opportunity cost/range of feasibility for new product W, determine shadow prices, assess whether ₹35 profit covers resource opportunity cost, and give definitive recommendation
- For all parts: Show complete working with formulas, degrees of freedom, and critical values from enclosed table; interpret results in business context

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	25%	12.5	All calculations mathematically accurate: correct slope (b≈8.93) and intercept (a≈82.71) for trend line; proper t-statistics for both independent (t≈1.89) and paired tests; correct LPP solution with optimal values (X=0, Y=150, Z=100) and profit ₹13,000; accurate shadow price calculation for new product evaluation	Minor computational errors in one sub-part (e.g., wrong degrees of freedom, arithmetic mistakes in LPP constraints) but correct formulas and approach; partial solutions with some correct intermediate steps	Major conceptual errors: wrong statistical test selection (z-test instead of t-test), incorrect LPP formulation (missing constraints or wrong objective), or fundamental misunderstanding of trend analysis methodology
Framework citation	15%	7.5	Explicitly states all formulas used: least squares (b=ΣXY/ΣX²), pooled variance t-test, paired t-test (d̄/(s_d/√n)), simplex method or graphical solution steps; references appropriate statistical tables and degrees of freedom calculations	Uses formulas implicitly without clear statement or has minor notational inconsistencies; mentions methods generally without specific formula presentation	No formulas shown, purely arithmetic approach without methodological framework; confuses frameworks (e.g., uses chi-square for mean comparison, simple averaging for trend)
Case / Indian example	15%	7.5	Contextualizes TATA Power as India's integrated power company with relevance to national grid planning; relates Techline Pvt. Ltd. scenario to Make in India/smart manufacturing initiatives; discusses how such analyses inform PSUs like NTPC or state electricity boards	Brief mention of Indian business context without deep integration; generic references to 'Indian industry' without specific sector application	Purely abstract numerical solution with no Indian context; treats TATA Power and Techline as fictional entities without real-world relevance
Multi-perspective analysis	25%	12.5	For (a): discusses limitations of linear trend (2023 outlier, COVID impact, need for non-linear models); for (b): compares independent vs paired designs, notes outlier in Feed B (33 kg), discusses Type I/II errors; for (c): analyzes sensitivity of solution, identifies binding constraints, discusses integer programming limitations, evaluates opportunity cost vs direct profit for Product W	Some analytical depth in one or two sub-parts but superficial treatment elsewhere; notes obvious limitations without exploring alternatives	Single perspective only—mechanical calculations without interpretation; no discussion of assumptions, limitations, or alternative analytical approaches
Conclusion & recommendation	20%	10	Clear definitive conclusions: 2025 forecast value with confidence caveat; explicit reject/fail-to-reject decisions with p-value interpretations; optimal production quantities (Y=150, Z=100, X=0); definitive recommendation on Product W with numerical justification (shadow price ₹10 for materials, ₹7.5 for labor, opportunity cost ₹35=₹35, indifferent or marginally reject); actionable managerial insights for TATA Power and Techline	Conclusions present but vague or inconsistent with calculations; recommendations generic without specific numerical backing; hedging without clear decision	Missing conclusions for sub-parts; conclusions contradict calculations; no recommendations or purely descriptive closing without decision-making

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